Second order Jacobi approximation with applications to fourth-order differential equations

Second order Jacobi approximation in non-uniformly weighted Sobolev space is investigated. Some approximation results on various orthogonal projections are established, which serve as the mathematical foundation of Jacobi spectral methods for differential equations of fourth order. Jacobi spectral schemes are provided for several model problems. The convergence is proved. Numerical results agree well with theoretical analysis and show the efficiency of this new approach.